Transversal homotopy theory

Jonathan Woolf

Implementing an idea due to John Baez and James Dolan we define new
invariants of Whitney stratified manifolds by considering the homotopy
theory of smooth transversal maps. To each Whitney stratified manifold we
assign transversal homotopy monoids, one for each natural number. The
assignment is functorial for a natural class of maps which we call
stratified normal submersions. When the stratification is trivial the
transversal homotopy monoids are isomorphic to the usual homotopy groups.
We compute some simple examples and explore the elementary properties of
these invariants.
We also assign `higher invariants', the transversal homotopy categories,
to each Whitney stratified manifold. These have a rich structure; they are
rigid monoidal categories for n > 1 and ribbon categories for n > 2. As an
example we show that the transversal homotopy categories of a sphere,
stratified by a point and its complement, are equivalent to categories of
framed tangles.